A Class Period on Spacetime-Smart 3-Vectors with Familiar Approximates Matt Wentzel-Long1, A) and P

A class period on spacetime-smart 3-vectors with familiar approximates Matt Wentzel-Long1, a) and P. Fraundorf1, b) Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA (Dated: 3 July 2019) Introductory physics students have Newton's laws drilled into their minds, but historically questions related to relativistic motion and accelerated frames have been avoided. With help from the metric equation, these 3- vector laws can be extended into the relativistic regime as long as one sticks with only one reference frame (to define position plus simultaneity), and considers something students are already quite familiar with, namely: motion using map-frame yardsticks as a function of time on clocks of the moving-object. The question here is: How may one class period in an intro-physics class, e.g. as a preview before kinematics or later during a day on relativity-related material, be used to put the material we teach into a spacetime-smart context? CONTENTS I. introduction 1 II. college physics table 1 III. introphysics in general 2 IV. cautions 3 Acknowledgments 4 A. possible course notes 4 1. spacetime version of Pythagoras' theorem 4 2. traveler-point kinematics 4 FIG. 1. Accelerometer data from a phone dropped and caught 3. dynamics in flat spacetime 5 three times. During the free-fall segments, the accelerometer 4. dynamics in curved spacetime 5 reading drops to zero because gravity is a geometric force (like centrifugal) which acts on every ounce of the phone's structure, and is hence not detected. The positive spikes occur I. INTRODUCTION when the fall is arrested as the falling phone is caught (also by hand) before it hit the floor. Consider driving a car: When you look at your speedometer, what are you seeing? The reported speed is not relative to some inertial frame with synchronized through the floor. It also fails to detect inertial forces, clocks on the side of the road, since speedometers use like those which push you back into the seat (or to the the rotation of the wheels1 (which make static contact outside of a curve) when your car accelerates (or follows with the road) per unit time on the car's on-board clocks. a curved path). This is good news, coming from general This \proper" ratio2 of map distance ∆x to traveler time relativity, which says that our accelerometer only detects ∆τ at any speed (e.g. even if lightspeed as for Mr. proper forces but that the \undetected" class of geomet- Tompkins3 was only ' 2:5 mph) turns out to be propor- ric forces (associated with accelerated frames or curved tional to 3-vector momentum ~p, to have no upper limit, spacetime) can in general be approximated locally as if and to also be most simply related to kinetic energy and they are one (or more) proper forces. This honored tradi- the time available for driver and pedestrian to react after tion, of treating geometric forces as proper, was of course the danger becomes apparent. It reduces to map distance started in the 17th century by none other than Issac New- ∆x per unit map time ∆t only at low speeds. ton himself. Another remarkable everyday example of the \traveler point" approach is the fact that your phone accelerometer cannot detect gravity, as shown in Fig. 1. It only II. COLLEGE PHYSICS TABLE detects the normal force which prevents us from falling For example, in our college algebra-based \basic physics II" class, Walker's text4 has a chapter near the end on relativity. Our strategy is to introduce the book's a)[email protected] tools, along with these traveler-point tools, for dealing b)[email protected] with problems of time dilation, unidirectional velocity 2 TABLE I. Newton at any unidirectional speed in (1+1)D, from (c∆τ)2 = (c∆t)2 − (∆x)2. Quantity# Variable! standard offering traveler-point version low-speed version time dilation γ ≡ ∆t=∆τ γ = 1=p1 − (v=c)2 γ = p1 + (w=c)2 γ ' 1 (vab+vbc) relative velocities v ≡ ∆x=∆t, w ≡ ∆x=∆τ vac = 2 wac = γacγbc(vab + vbc) vac ' vab + vbc (1+vabvbc=c ) momentum p = mγv p = mw p ' mv total energy E = γmc2 E = γmc2 E ' mc2 2 2 1 2 kinetic energy K = (γ − 1)mc K = (γ − 1)mc K ' 2 mv addition, and relativistic energies/momenta. Length contraction is off the table, because it requires two extended frames with synchronized clocks. Proper-velocity ~w = γ~v and rest-mass m is used instead of \relativistic mass" to preserve the standard relationship between momentum and velocity, and students are only being asked to mas- ter that subset of problems posed in the book which can solved with or without these \hybrid kinematic" tools, as shown Table I. III. INTROPHYSICS IN GENERAL FIG. 2. Two 6:5 TeV LHC protons send messages to each other, while passing at proper velocities of about ' 6929 More generally we suggest initial mention (even if only lightyears/traveler-year, for a collider energy advantage of in passing) of the \traveler-point variables" (chosen be- Krel=K ' 13; 859 times the energy of stationary target colli- cause they either have frame-invariant magnitudes or be- sion. cause they don't require synchronized clocks), namely traveler or proper time τ, proper velocity defined as map distance per unit traveler time ~w ≡ ∆~x=∆τ, and the net proper force Σξ~ = m~α felt by on-board accelerometers. These are approximated at low speeds by the more familiar map time t, coordinate velocity ~v ≡ ∆~x=∆t, and net map-based force Σf~ ' ∆~p=∆t. By sticking with displacements ∆~x and simultaneity defined by a single bookkeeper or map reference frame (i.e. the metric), as shown in Table II we can simply describe time-dilation γ ≡ ∆t=∆τ and constant unidirectional proper acceleration ~α at any speed, even when there's no time to explore 3-vector proper velocity/acceleration or multi-frame phe- nomena like length contraction. The unidirectional proper-velocity addition equation given in Table II, for example, allows students to see the advantage of colliders over accelerators in more vis- ceral terms, which may even fire up the imagination of NASCAR fans (think of land and relative speed records for particles) as depicted in the relative velocity illustra- tion of Figure 2 (inspired by an XKCD cartoon). Simi- FIG. 3. Round trip times and a sample thrust profile for a larly the unidirectional equations of constant proper ac- spaceship capable of constant 1-gee acceleration and avoiding celeration given in Table II allow students to easily cal- collision with atoms. culate the map and traveler times elapsed on constant proper-acceleration round trips between stars, as illus- trated in Figure 3. is K = (γ−1)mc2. Remarkably, however, in curved space In passing, we should also mention the curious relation- time and in accelerated frames, relations like this also ex- ship between various energies and the time-dilation or press potential energies associated with geometric forces. differential-aging factor γ ≡ ∆t=∆τ. The basic relation- This is easiest to see when standing in an artificial (cen- ship, given in Table II, allows us to say that kinetic energy trifugal) gravity well, where the rotational kinetic energy of motion with respect to inertial frames in flat spacetime from a fixed external point of view looks like a potential 3 TABLE II. Traveler-point dynamics in flat (1+1)D spacetime: Conserved quantities energy E = γmc2 and momentum ~p = m ~w, where differential-aging factor γ ≡ δt/δτ, proper velocity ~w ≡ δ~r/δτ ≡ γ~v, coordinate acceleration ~a ≡ δ~v/δt. In the first 4 rows, τ is traveler time from \rest" with respect to the map frame, and α is a fixed space-like proper acceleration vector. Asterisk means that the (1+1)D relation also works in (3+1)D. relation w c (1+1)D c ατ map time elapsed t t ' τ t = α sinh[ c ] 1 2 c2 ατ map displacement ~x ~x ' ~vot + 2~at x = α (cosh[ c ] − 1 ) 1 2 ατ aging factor γ ≡ δt/δτ γ ' 1 + 2 (v=c) γ = cosh[ c ] ατ proper velocity ~w ≡ δ~x/δτ ~w ' ~v ' ~vo + ~at w = c sinh[ c ] *momentum ~p ~p ' m~v ~p = m ~w = m(γ~v) 2 1 2 2 *energy E E ' mc + 2 mv E = γmc ~ ~ felt (ξ) $ map-based (f) f~ ' ξ~ f~ = ξ~ force conversions *work-energy δE ' Σf~ · δ~x δE = Σf~ · δ~x *action-reaction f~AB = −f~BA f~AB = −f~BA *map-force:momentum Σf~ = δ~p/δt ' m~a Σf~ = δ~p/δt ;Σξ~ = m~α *felt force:acceleration 1 2 2 energy well of depth U ' 2 m! r to the rotating inhabi- usefulness in the case when there are \oppositely tant. However, it also turns out to be true in a spaceship charged" force-carriers is behind the 19th cen- of length L undergoing constant proper acceleration α, tury distinction between magnetic and electrostatic where the energy to climb from trailing to leading end is fields. 2 ∆U = (∆tleading=∆ttrailing − 1)mc ' mαL, and in the gravity of a non-spinning sphere of mass M and radius • 4th caution: The simultaneity of separately located R, where the escape energy for mass m on the surface events is also frame dependent, i.e. differently mov- (when R is much more than the Schwarzschild radius) is ing observers may disagree on which of two \space- 2 like separated" events came first, just as the filial Wesc = (∆tfar=∆τ −1)mc ' GMm=R, where tfar is time elapsed on the clocks of distance observers.

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